Every solid is a flat shape that learned to fold.

Three little words: face, edge, vertex.

Before we fold anything, you need three words. They're the building blocks mathematicians use to describe any solid, so once you've got them, you can describe a cube, a Toblerone box, or the Great Pyramid with the same handful of terms.

Here's a sentence to keep: faces are the flat bits, edges are the lines, vertices are the points. A cube has 6 faces, 12 edges, and 8 vertices — and a little later you'll discover those three numbers are secretly related by a rule that works for almost every solid you'll ever meet.

One thing trips almost everyone up at first: counting. It's easy to under-count edges and vertices because you can only see the front of a solid — the back faces, edges and corners are hiding. The fix is to be systematic. Count the top, then the bottom, then the bits in between; or better still, unfold the solid into its net so every face is lying flat in front of you, then add up. That's the whole reason nets are about to become your best friend: counting a flat picture is far less dizzying than counting a 3D object you keep having to rotate in your head.

The solids you'll see again and again.

Most of geometry's solids fall into a few families, and once you know the families you can predict almost everything about a new shape. A prism is a solid with two identical ends joined by flat rectangles — slice it anywhere along its length and you get the exact same shape, like cutting a loaf of bread. The name even tells you the ends: a triangular prism has triangle ends, a hexagonal prism has six-sided ends. A pyramid has one base and triangular sides that climb to a single point at the top, called the apex; name the base and you've named the pyramid. And then there are the curvy ones — cylinders, cones and spheres — which trade some flat faces for smooth, rounded surfaces.

That curvy/flat split matters more than it looks. A solid whose faces are all flat polygons is called a polyhedron (plural: polyhedra) — cubes, prisms and pyramids are all polyhedra. Cylinders, cones and spheres are not, because part of their surface is curved. Keep that distinction in your back pocket; it's going to explain a beautiful pattern near the end of the page.

So what exactly is a net?

A net is a 2D shape that folds up — along its edges, with no gaps and no overlaps — into a 3D solid. Imagine taking a cardboard box and very carefully snipping along some of its edges until the whole thing flops open flat on the table. What you'd be left with is the box's net: every face laid out side by side, still joined at the edges that didn't get cut.

This is genuinely useful, not just a party trick. The net shows you everything the solid is made of, all at once and lying flat where you can see it. You can count the faces. You can spot which faces are the same size. And — coming up — you can add up their areas to find out how much wrapping paper or cardboard the solid needs.

Two quiet rules make a shape a true net. First, no gaps: when you fold it up, the edges must meet so the solid is fully sealed, with no missing face. Second, no overlaps: no two faces are allowed to land on top of each other. Break either rule and your flat shape might be a pretty pattern, but it isn't a net. Here's the surprising part, though — a single solid usually has many different nets, because there are lots of valid ways to cut it open. A cube has eleven of them, which we'll explore shortly. The cube's most famous net is a cross: a column of four squares with one square sticking out on each side. Fold the four arms up around the middle square and the sixth square drops on top like a lid. Let's actually do it.

Fold a flat net up into a solid.

Drag the slider from 0% (flat net) to 100% (closed solid) and watch the flat shape lift off the page and seal itself into a 3D shape. Or tap Fold up and let it do it for you. Switch between the cube and a square pyramid to see two different nets close.

Did you catch the trick? Every square in the net becomes exactly one face of the solid, and every cut edge gets zipped back to a partner. Nothing is added and nothing is thrown away — folding just rearranges the same flat pieces in space. That's why the net is such a faithful blueprint: the solid is the net, only standing up.

Watch the cube closely as you scrub the slider. The middle square stays put as the floor; the four arms swing upward to become the walls, and the very last square — the one at the far end of the long strip — flips over to land as the lid. Notice how each fold happens along an edge that was already there in the flat net. Then try the pyramid: this time the four triangles don't just stand up to ninety degrees, they keep leaning inward until all four tips kiss at a single point — the apex. Same idea, different choreography. The flat shape always knows exactly where every piece is supposed to go.

Different solids, different nets.

Once you trust the idea, you can predict the net of almost anything by asking one question: what flat pieces would I need, and how do they join?

That cylinder fact is a lovely one to hold onto. Peel the label off a tin of soup without tearing it and you get a rectangle — and its long side is exactly the distance around the tin. The net doesn't lie. (You already folded the pyramid above; flip back and watch those four triangles rise to a single point.)

Prisms all follow the same recipe, which makes their nets easy to draw from memory: take the two end shapes, then lay a row of rectangles between them — one rectangle for each side of the end shape. A triangle has three sides, so a triangular prism gets three rectangles; a pentagon has five, so a pentagonal prism gets five. Cones are the odd one out and the most fun to picture: snip a cone from its tip down to the rim and roll it open, and the curved surface flattens into a sector — a slice of pie — while the base stays a circle. Roll the slice back up and the point of the slice becomes the point of the cone. Every solid you meet has a net waiting inside it; you just have to imagine where to make the cuts.

Not every six-square shape is a cube net.

Here's where it gets sneaky. A cube net always has six squares — but six squares stuck together do not automatically fold into a cube. Some leave a gap. Some make two faces overlap. It turns out there are exactly 11 different nets that fold into a cube — out of the 35 ways you can join six squares edge-to-edge, only 11 actually work.

You don't have to memorise all eleven (though it's a great rainy-afternoon challenge to find them yourself). You just need a feel for what makes one work. A good cube net is spread out: the squares wander around so that, as they fold, each one has empty space to swing into. The shapes that fail are the cramped ones, where squares are packed so tightly that two of them want to occupy the same spot once you start folding. Keep that picture in mind as you try the sorter below.

Below are six arrangements. For each one, decide: does it fold into a cube, or not? Tap your answer to check.

Count the parts — and find the hidden pattern.

Now that a solid is something you can unfold and inspect, counting its parts is easy. Tap a solid to see its faces (F), edges (E) and vertices (V) — then watch what happens when you put them together.

For every polyhedron — a solid whose faces are all flat — there's an astonishing pattern spotted by the mathematician Leonhard Euler nearly 300 years ago:

F − E + V = 2

Faces, minus edges, plus vertices, always equals two. A cube: 6 − 12 + 8 = 2. A triangular prism: 5 − 9 + 6 = 2. A pyramid, a tetrahedron, a soccer ball's flat-faced cousin — all the way to two, every single time. The curvy solids (cylinder, cone, sphere) bend the rules because they aren't built from flat faces, so don't expect them to land on 2 — but for anything with proper flat faces and straight edges, that little equation is rock solid.

What makes this magical is how stubborn it is. You could invent a wild new polyhedron right now — squash a corner, slice off an edge, glue two prisms together — and as long as it stays one solid lump with flat faces and no holes punched through it, F − E + V will still come out to exactly 2. Cut a face into two and you also add an edge and sometimes a vertex; the changes quietly cancel out and the total holds. Mathematicians love patterns that refuse to break, and this one has held up for centuries. Try to find a polyhedron that breaks it — gently, you won't manage, and the failing to break it is exactly the point.

A net is a shortcut to surface area.

Here's the real pay-off. Surface area is the total amount of "skin" on a solid — how much paint, foil or wrapping it would take to cover it completely. On a finished 3D shape that's awkward to picture, because the faces point in all directions. But unfold it into a net and every face lies flat in front of you, side by side.

Now it's simple: find the area of each flat piece, then add them up. For a cube with edges of length 4 cm, every one of the six square faces is 4 × 4 = 16 cm², so the surface area is 6 × 16 = 96 cm². No 3D wizardry needed — the net turned a solid problem into six flat ones.

The same trick rescues you on trickier solids. A cuboid that's 5 cm by 3 cm by 2 cm has its faces in matching pairs: two of 5 × 3, two of 5 × 2, and two of 3 × 2. Lay the net out, work out the six rectangles, add them, and you're done — no need to wrestle the box around in your imagination. This is exactly the move designers make when they work out how much cardboard a box will cost, or how much glaze covers a tile: unfold, measure flat, add up. Surface area stops being scary the moment you stop trying to do it in 3D.

You're surrounded by nets.

This isn't a maths-classroom-only idea — it's how almost every box you own was made. A cereal box, a juice carton, a tube of sweets, a pizza box, a perfume case: each one began as a single flat sheet of card, printed all over while it was still flat (much easier than printing on a finished box), then cut, scored along the fold lines, folded up, and glued. The flat shape the machine starts with is a net.

Try it at home: very carefully unstick a small empty cardboard box at its glued seam and open it out flat. You'll be holding its net — and if you look closely you can usually spot the extra glue tabs the designers added so the edges have something to stick to. Engineers, packaging designers, and even people who fold origami all think in nets, turning flat sheets into solid shapes and back again.

The idea reaches further than boxes, too. A flat map of the round Earth is a net problem in reverse — you can't peel a sphere into a flat sheet without stretching or tearing it, which is exactly why every world map distorts something. Architects build cardboard nets to preview buildings before pouring concrete; video-game artists "unwrap" a 3D character into a flat net so they can paint its skin on a 2D image; sheet-metal workers cut flat nets that get bent into ducts and machine parts. Wherever something flat has to become something solid — or the other way round — nets are doing the quiet, clever work behind the scenes.